What is the integral of ((x^3-x^2)^2)/(sqrt(x)) ?

The integral of ((x^3-x^2)^2)/(sqrt(x)) is 2/13 x^{13/2}-4/11 x^{11/2}+2/9 x^{9/2}+C

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integral of ((x^3-x^2)^2)/(sqrt(x))

\int \frac{( x ^{3} - x ^{2} ) ^{2}}{x} d x

\frac{2}{13} x^{\frac{13}{2}} - \frac{4}{11} x^{\frac{11}{2}} + \frac{2}{9} x^{\frac{9}{2}} + C

Solution steps

\int \frac{( x ^{3} - x ^{2} ) ^{2}}{x} d x

Expand

\frac{( x ^{3} - x ^{2} ) ^{2}}{x} : x^{\frac{11}{2}} - 2 x^{\frac{9}{2}} + x^{\frac{7}{2}}

= \int x^{\frac{11}{2}} - 2 x^{\frac{9}{2}} + x^{\frac{7}{2}} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int x^{\frac{11}{2}} d x - \int 2 x^{\frac{9}{2}} d x + \int x^{\frac{7}{2}} d x

\int x^{\frac{11}{2}} d x = \frac{2}{13} x^{\frac{13}{2}}

\int 2 x^{\frac{9}{2}} d x = \frac{4}{11} x^{\frac{11}{2}}

\int x^{\frac{7}{2}} d x = \frac{2}{9} x^{\frac{9}{2}}

= \frac{2}{13} x^{\frac{13}{2}} - \frac{4}{11} x^{\frac{11}{2}} + \frac{2}{9} x^{\frac{9}{2}}

Add a constant to the solution

= \frac{2}{13} x^{\frac{13}{2}} - \frac{4}{11} x^{\frac{11}{2}} + \frac{2}{9} x^{\frac{9}{2}} + C

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What is the integral of ((x^3-x^2)^2)/(sqrt(x)) ?
The integral of ((x^3-x^2)^2)/(sqrt(x)) is 2/13 x^{13/2}-4/11 x^{11/2}+2/9 x^{9/2}+C